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Fourier-Transform Infrared Spectroscopy(FTIR)
Principles of the Fourier Transform Spectroscopy
FTIR Spectrometer
Mostly based on:
Introductory Fourier Transform Spectroscopy Robert John Bell &
Michelson Interferometer
rktEE cos0
c
k
22
1~
xtEE ~2cos0
101~2cos xtEtrE
X1/2
X1/2
1
2
(r,t)
102~2cos xtEtrE
~cos~2cos2 1021 xtEtrEEER
~2cos1~)( Bi
Polychromatic source
~~2cos~~~~~2cos1~)(maxmaxmax
~
0
~
0
~
0
dBdBdBi
max
~
0
~~2)0(
dBi 02
1~~)(max
~
0
idBi
3.0
2.5
2.0
1.5
1.0
0.5
0.0
i
-3 -2 -1 0 1 2 3
i
i
~~2cos~)0(2
1)()()(
max~
0
dBiiii
Consider a source of monochromatic wave:
2
Ei time
iThe detector measures where
0
Fourier-Transform
~~2cos~)0(2
1)()()(
max~
0
dBiiii diiB ~2cos)0(2
1)(~
0
FT
max
Two obvious technical difficulties:
1. The retardation is always finite, max
Consequence: Signal (in frequency space) appears convoluted with an Instrumental Line Shape (ILS) function Limits the measuring resolution
Solution: Apodization
2. The zero retardation =0 (ZPD) is needed for calculating the spectrumConsequence: Asymmetry of the interferogram, phase error
Solution: Double-sided interferogram
~)()(~22 ~
BfBtVolttii scannervscanner VfFFTDetectortV
The FTIR spectrometer:
Bruker 113v/IR98
Advantages of the FTIR spectrometer
mmrc.caltech.edu/FTIR/FTIRintro.pdf
http://www.thermo.com/eThermo/CMA/PDFs/Product/productPDF_21615.pdf
Important Properties of Fourier-Transform
dxexfg xi
~2)(~
~~)(~2 degxf xi
dxxxfidxxxfg ~2sin)(~2cos)(~
~~2sin~~~2cos~)( dxgidxgxf
1. If f(x) is even (f(x)=f(-x) ) FT(f(x)) is real and even
If f(x) is odd (f(x)=-f(-x) ) FT(f(x)) is imaginary and odd
http://dagsaw.sdsu.edu
Convolution: dtgftgtf
)(
2. Convolution Theorem: The FT of the product of two function is the convolution of their individual FTs
)()()()(
)()(
)()(tgFTtfFTtgtfFT
tgFTtg
tfFTtf
~2cos~)( BiFT
max
diB
~2cos~
-max
)()()(~2cos~
iFTdiB
(x)=1 if -max max
0 if > max
Finite retardation Multiplication of the real space interferogram with a “boxcar function”
4
3
2
1
0
-4 -2 0 2 4
1.0
0.0-4.0-3.5-3.0-2.5-2.0-1.5-1.0-0.50.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
4
3
2
1
0
-1
Am
plit
.
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0Freq.
11~2cos~)( Bi
FT )~(~2cos~
111
diB
)~(~)(~2cos~11 GBdiB
Finite retardation the real spectrum is the “true spectrum” B(1) convoluted with the ILS
Monochromatic radiation of frequency 1 :
(x)=1 if -max max
0 if > max
FT maxmax
~2sin2)~( cG
1.0
0.0
-5 -4 -3 -2 -1 0 1 2 3 4 5
4
3
2
1
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
max)
max)
1=3
Instrumental Line Shape (ILS)
1.0
0.5
0.0
-5 -4 -3 -2 -1 0 1 2 3 4 5
2max
1/max
0.605/max
max=0.5
ILS
Broadening of the spectral line
max11max~~sin~2~ cBB
Delta Function
4
3
2
1
0
5.04.54.03.53.02.52.0
The resolution of an instrument:
Two adjacent lines of equal intensity, having sinc2(x) shape, are resolved when the center of one line coincides with the first zero of the other
1 Rayleigh criterion :
Intuitively, the resolution min=1/max
2. FWHH criterion: Two triangularly shaped lines of equal intensity are resolved when the spacing between lines is grater than the FWHH of either line.
6
5
4
3
2
1
0
-1
6543210
0.5/max
1.0
0.0
6543210
6
5
4
3
2
1
0
-1
6543210
0.73/max
1.0
0.0
6543210
Apodization
If we use the triangular function (x) instead of the boxcar function (x) ILS sinc2(x)
2.0
1.0
0.0
-4 -3 -2 -1 0 1 2 3 4
1.0
0.0
-2 -1 0 1 2
1.0
0.0
-4 -3 -2 -1 0 1 2 3 4
max
0.88/max
2/max
1.0
0.5
0.0
-5 -4 -3 -2 -1 0 1 2 3 4 5
2max
1/max
0.605/max
Other Apodization Functions:
Particularly common in FTIR spectrometers:
Happ-Genzel
Sampling Phase Errors & Two-Side Interferogram
diiB ~2cos)()(~
0
Phase errors may arrive from misalignments of the interferometer (optical effects), from filters (electronic effects) and/or from sampling errors
Let’s assume that the ZPD is not correctly determined: 0 )()( ii
diiBmeas~2cos)()(~
0
The effect: ~2exp~~ iBBmeas
Consider the complex FT on both sides of =0, from -max to max :
diiiBmeas~2exp)()(~
Compute both real (cos) part, C(), and imaginary (sine) part, S() of the FT:
~~~~ 22 BSCBmeas
Mathematical proof for phase correction : TO BE ADDED
Sampling interval
Nyquist Criterion: The sampling frequency (wavenumber) should be equal or larger than twice the highest measured frequency (wavenumber).
The effect of undersampling is aliasing, or folding:
max2 ffsamp
http://zone.ni.com/devzone/cda/tut/p/id/5509
ff
ffffff
fsamp
realsampimagsamp
samp
real 22
alias
FTIR Spectrometer:
~2cos~)( Bi
~)(~22 ~
BfBtii scannervscanner VfFFTtV
tVBti scanner 2~2cos~)(
tfEtEtE 2coscos)( 00
~2~ scannerVf
cms
cm
sHz
11
It is better to sample equal intervals of retardation () rather than equal intervals of time (t)
If we want to measure up to max~
max
maxmaxmaxmax ~22
1~22~2~
scanner
sampscannersamp
Nyquist
scannerV
tVfVf
1
max
2
max~2
1~22
1
cmcm
Vt samp
tV
scanner
sampscanner
The Scanning Process & Duty Cycle Efficiency
Vsc
ann
er
time
Duty Cycle Efficiency = cycle wholea of Time
measured is raminterferog whileTime
When the interferograms are sampled coherently, i.e. between same retardation limits around ZPD:
)(N Scans ofNumber Signal scans
)(N Scans ofNumber Noise scans
The noise varies continuously (randomly) with time, it is not expected to have the scanning cycle periodicity
scansNSNR
Different “flavors” of scanning cycles:
Rapid-Scan Interferometer
Step-Scan Interferometer
Requirements:
Data collection needs to start at the same retardation from the ZPD each cycle
Sampling needs to take place at the same retardations each cycle
tVscanner
2
http://www.chem.uic.edu/tak/chem524/notes16/figureIR_3.gif
~)(~22 ~
BfBtii scannervscanner VfFFTtV
How the interferogram is sampled or why we need He-Ne laser and white light ?
The He-Ne frequency sets the sampling frequency (for one sample/cycle):
115800~~
cmsampNeHe 1
maxmax 7900~5.0~ cm
The white light determines when the first data point is taken (t=0), i.e. initiates data collection
Vtrig
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-4 -2 0 2 4
Initiate Data Collection
White Light
He-Ne
IR
time
Sub-sequential data collection is triggered by the zero crossings of the He-Ne laser interferogram
NeHescannerNeHe VHzf ~2
Example: 13.02 scmVscanner
KHzf NeHe 74.4158003.0 (This is how OPUS labels scanner velocity)
Vtrig
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-4 -2 0 2 4
Initiate Data Collection
White Light
He-Ne
IR
time
The sampling electronics (ADC board) starts measuring when a thresold signal Vtrig is detected
It is important that the white light interferogram occurs before the IR interferogram
DAQ can only take place when scanning mirror moves in one direction single-sided interferogram and poor duty cycle efficiency
Position of the white light interferogram must remain constant and cannot be adjusted with maximum retardation
How is the phase correction achieved? – To be discussed
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